I would mow first, then aerate. That way you won't have a plug with long hair. I was talking with one of my customers today about aerating and she asked me the same question. I don't know about you but I would wait to aerate after your last mow, that way you won't be running over plugs. Thats what I'm going to do.
You should aerate while the grass is actively growing. I would mow first and them aerate. This way the plugs have a week to break down before mowing. You mentioned that you don't mow this one all the time, so mow first if you have this option.